Math 105 Course Outline
Week 11 Goals
Overview
In first-semester calculus, you learned what it meant to talk about the limit
of a function. We begin this week by discussing what it means to talk about
the limit of an infinite list of numbers (which we call an infinite sequence). For
1
example, the sequence 1, 21 , 41 , 18 , 16
, . . . converges to zero while both of the sequences 0, 1, 2, 3, . . . and −1, 1, −1, 1, . . . fail to tend to a limit (in other words,
diverge).
The next question we turn to is: Given an infinite list of numbers, what does
it mean to add them all up? (A sum of an infinite list of numbers is called
an infinite series.) We discuss certain special types of infinite series (such as
geometric series, telescoping series, and p-series), as well as tests that allow
you to determine convergence and divergence for very general series.
Learning Objectives
These should be considered a minimum, rather than a comprehensive, set of
objectives.
By the end of the week, having participated in lectures, worked through the
indicated sections of the textbook and other resources, and done the suggested
problems, you should be able to independently achieve all of the objectives
listed below.
1
Ref
11–01
Learning Objective
Sequences: Basic properties
Objective 1: Give the definition of a sequence {an }. See p. 527.
[Conceptual]
Objective 2: Compute the first several terms of a sequence given
to you either by an explicit formula or by a recursive definition.
[Procedural]
Example problem: If an+1 = 3an − 1 and a1 = 1, find all of a1 , a2 ,
a3 , and a4 .
Reading: Text §8.1 (pp. 526 – 528)
Practice problems: Text pp. 534–535: 9–16.
Objective 3: Explain what it means for a sequence to have a limit
(converge) or diverge. Given several terms of a sequence, be able
to formulate a guess as to its limit. [Conceptual + Procedural]
You do not have to understand the formal definition of the limit
of a sequence presented on pp. 545–546.
Reading: Text §8.1 (p. 529)
Practice problems: Text p. 535: 23-30.
Objective 4: [Procedural] Know how to compute limits of sequences using either of the following methods:
• Writing the nth term as f (n), where f is a function for which
limx→∞ f (x) exists. See Theorem 8.1 on p. 537.
• Applying the squeeze theorem. See §8.2 (p. 542).
Example problem: If an = en /(1 + 3en ), what is limn→∞ an ?
Example problem: Find the limit of the sequence an =
Reading: Text §8.2 (pp. 537, 542)
Practice problems: Text p. 546: 9-26, 43–46.
n2 +n sin(n)
.
n3 +1
Ref
11–02
Learning Objective
Sequences: Continued
Objective 1: Apply the properties of limits summarized in Theorem 8.2 on p. 538. [Procedural]
Example problem: If an and bn are two sequences with
limn→∞ an = −3 and limn→∞ bn = 10, find limn→∞ (3an − 7bn ).
Practice problems: Text p. 547: 63, 67, 69.
Objective 2: Recognize when a sequence is increasing, decreasing, bounded, or monotone. [Conceptual]
Objective 3: Be able to state Theorem 8.5: A bounded monotonic
sequence converges. Recognize (using the skills acquired in Objective 2) when this result applies. [Conceptual]
Example problem: Which of the adjectives increasing, decreasing,
bounded apply to the sequence whose nth term is an = (−1)n /n?
n
?
What about an = n2 ? What about an = n+1
Reading: Text §8.2 (pp. 539–540)
Practice problems: Text p. 546: 1–4.
Objective 4: Recognize examples of geometric sequences and determine whether they converge or diverge. [Conceptual and Procedural]
Example problem: Does the sequence {(0.95)n } converge or diverge?
Reading: Text §8.2 (pp. 540–541)
Practice problems: Text p. 546: 35–42.
Ref
11–03
Learning Objective
Infinite Series
P∞
Objective 1: Give the definition of an infinite series k=1 ak and
explain what is meant by the sequence of partial sums. Relate the
convergence or divergence of the series to the sequence of partial
sums. [Conceptual]
Example
problem: Compute the first four partial sums of the series
P∞
n
(−1)
/n.
n=1
Example
Find a formula for the nth partial sum of the
P∞problem:
1
series k=1 k(k+1)
. Use this to determine the value of the infinite
series.
Reading: Text §8.1 (pp. 532–534, 552–553)
Practice problems: Text pp. 534–535: 6, 7, 8, 60–67 and pp. 553–554:
47–58.
Objective 2: Recognize when a geometric series converges and be
able to compute its sum in that case. [Conceptual + Procedural]
P∞
Example problem: Does the series k=1 (−0.95)k converge or diverge? If it converges, what is its value?
Reading: Text §8.3 (pp. 550–552)
Practice problems: Text pp. 553–554: 7–46, 59.
Objective 3: Be able to manipulate convergent series according to
the rules described in Theorem 8.8 on p. 557. [Procedural]
P∞
k
Example problem: Evaluate the infinite series k=1 3−4
.
5k
Reading: Text §8.3 (pp. 557–559)
Practice problems: Text p. 567: 9–14.
Objective 4: [Procedural] Be able to use both of the following tests:
• The divergence test (Theorem 8.9, p. 559), which gives a sufficient condition for a series to diverge.
• The integral test (Theorem 8.11, p. 561), which shows the
equivalence between the convergence of a series and that of
an associated integral.
√
P∞
k
Example problem: Explain why the series k=1 √k+1
diverges.
P∞
1
Example problem: P
Does the series
k=2 k ln k converge or di∞
1
verge? What about k=1 √
?
5
k
Reading: Text §8.4 (pp. 559–564)
Practice problems: Text p. 567: 15–30, 44–49.